What, in your understanding, does due process entail? Because all a law enforcement officer needs to detain a person is "reasonable articulable suspicion." And all a law enforcement officer needs to arrest you (i.e. take you into custody) is "probable cause" to believe you committed the crime for which you're being arrested. The officer can be wrong, so long as he/she acts in good faith belief.
Due process involves the right to a hearing, where the person who was arrested/charged with a crime has an opportunity to require proof.
This particular case involves some unusual facts. The guy they arrested was born in the USA, but lived nearly his entire life in Mexico. Does he have a SS#? Did he ever attend US schools? Does he have a driver's license? Does he hold a legit job (which would require the SS#) or was he working under the table? If he has basically no footprint of living as a citizen, it's not really shocking that he was suspected of being undocumented. That birth certificate would be "on record" only in the county where he was born.